Uncertainty amplification due to density/refractive index gradients in background-oriented schlieren experiments

Abstract

We theoretically analyze the effect of density/refractive-index gradients on the measurement precision of Volumetric Particle Tracking Velocimetry (V-PTV) and Background Oriented Schlieren (BOS) experiments by deriving the Cramer-Rao lower bound (CRLB) for the 2D centroid estimation process. A model is derived for the diffraction limited image of a particle or dot viewed through a medium containing density gradients that includes the effect of various experimental parameters such as the particle depth, viewing direction and f-number. Using the model we show that non-linearities in the density gradient field lead to blurring of the particle/dot image. This blurring amplifies the effect of image noise on the centroid estimation process, leading to an increase in the CRLB and a decrease in the measurement precision. The ratio of position uncertainties of a dot in the reference and gradient images is a function of the ratio of the dot diameters and dot intensities. We term this parameter the Amplification Ratio (AR), and we propose a methodology for estimating the position uncertainties in tracking-based BOS measurements. The theoretical predictions of the particle/dot position estimation variance from the CRLB are compared to ray tracing simulations with good agreement. The uncertainty amplification is also demonstrated on experimental BOS images of flow induced by a spark discharge, where we show that regions of high amplification ratio correspond to regions of density gradients. This analysis elucidates the dependence of the position error on density and refractiveindex gradient induced distortion parameters, provides a methodology for accounting its effect on uncertainty quantification and provides a framework for optimizing experiment design.

Uncertainty-based weighted least squares density integration for background-oriented schlieren

“…Work is ongoing to develop a methodology to estimate the covariance in displacement estimation between neighboring grid points from the cross-correlation plane for PIV/BOS measurements. Work is also ongoing on estimating the displacement uncertainty for tracking-based BOS measurements using the ratio of dot diameters in the reference and gradient images [10], and this is another avenue for the application of WLS based density integration.…”

“…al. [9], [10] made significant advancements in uncertainty quantification methods for BOS measurements by developing a method to report local, instantaneous, aposteriori density uncertainty across all points in the field of view. There has been a recent development in uncertainty quantification method for BOS measurements by Rajendran et.…”

“…al. [9], [10] to report local, instantaneous, a-posteriori density uncertainty across all points in the field of view. To achieve this, PIV-based displacement uncertainty quantification methods are used to estimate displacement uncertainties from cross-correlation BOS and propagated through the density integration procedure.…”

Practical Aspects of Designing Background-oriented Schlieren (BOS) Experiments for Vortex Measurements

“…The general implications discussed in different publications (Goldhahn and Seume 2007;Hargather and Settles 2012;Gojani et al 2013;Raffel 2015;Rajendran et al 2020) are that the sensitivity can be increased by increasing the focal length and by positioning the object closer to the camera (Raffel 2015;Hargather and Settles 2012). It is typically stated that the amount to which the object can be moved closer to the camera is limited by the need to keep the object in reasonable focus.…”

Meta-uncertainty for particle image velocimetry

“…There has also been recent progress in uncertainty quantification in Background Oriented Schlieren (BOS), an image-based density measurement technique. Developments include estimation of displacement uncertainty from dot tracking based processing [8,9], propagation of both tracking/cross-correlation based displacement uncertainties through the density integration chain [10], and utilizing the uncertainties to improve the density integration process by weighted least squares minimization [11].…”